Finding the equations of lines through a point that are tangent to a function

Question

What are the equations of two lines that intersect at (4,7) and are tangents to y=x^2?

Answer 1

The tangent's slope is given by y' = 2x.

Let (a,a²) be any point at which there's a tangent; then m = 2a.

And (4,7) is on that line, albeit (4,7) isn't the point of tangency.

 

So m = rise/run = (a²-7)/(a-4) = 2a....solving for a, we get 0 = a² - 8a + 7; 0 = (a-7)(a-1).

The x coordinates of the points of tangency are {1, 7}.  

The corresponding y coordinates are {1, 49}.

 

Therefore one tangent line's equation, at (1,1) is

y-1 = 2(x-1).

I leave it to you to write the other tangent line's equation.